You will get lost; trust me. The downtown Madison  spreads along a narrow isthmus between two large lakes, which doesn’t bode well for traffic (though you’re always within eight blocks of lakeside prettiness, I guess).
Given the topography, traffic can be maddeningly circuitous. Always keep an eye on the Capitol. Also keep in mind that east and west in street names are approximations—it’s actually closer to northeast and southwest. All east-west streets use the Capitol as the dividing point.
The main thoroughfare down the throat of the isthmus is East Washington Avenue (U.S. 151); it leads directly to, and then around, the massive state Capitol, which is connected to the university by State Street .
The main artery between east and west Madison is the white-knuckled swells of the Beltline—U.S. 12/14 and 151/18 (or any combination thereof). It’s unlikely you’ll be able to avoid the Beltline altogether; just avoid rush-hour peaks. (A Windy City bro-in-law has said of it, "The only place where Madison traffic ever rivals Chicago's.")
Like everywhere else, parking sucks here. Either not enough or expensive; worse, parking sentinels are thought to be omniscient.
Unlike Milkwaukee, nobody—drivers, bikers, pedestrians—follows the laws (but we claim we do). Pay attention out there!